Transcript lecture_10_philosophy

Lecture 10
Categorical Logic
Categorical statements
Why do we need Categorical Logic?
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Propositional logic does not cover all valid
logical forms.
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It is an important part of logic, but not the only part
of logic.
Is the following argument valid?
All human beings are mortal;
 Socrates is a human being;
 So, Socrates is mortal
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This is a valid argument. Can it be explained by
propositional logic? Not quite.
Argument:
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All human beings are mortal;
Socrates is a human being;
So, Socrates is mortal
1.
2.
3.
Analysis (Modus Ponens?):
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a)
b)
c)
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If Socrates is a human being, then Socrates is mortal.
Socrates is a human being;
So, Socrates is mortal
But statement 1 and statement (a) are not quite the
same thing.
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Another argument
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Some four-legged
creatures are gnus.
All gnus are herbivores.
Therefore, some fourlegged creatures are
herbivores.
Is this argument valid?
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Argument:
Some four-legged creatures are gnus.
 All gnus are herbivores.
 Therefore, some four-legged creatures are
herbivores.
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This argument is valid, but it cannot be
explained by propositional logic.
We need some kind of new logic.
Categorical Logic
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Categorical logic was
discovered long long ago
by Aristotle (384-322
BCE).
The arguments we have
discussed are called
syllogism.
Aristotle discovered all
valid forms of syllogism.
Categorical Logic
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Categorical logic studies inference between categorical
statements.
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It is the focus of today’s lecture to explain the structure of
categorical statements.
Typical inferences between categorical statements:
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Syllogism:
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Latin Square:
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Case: All As are Bs, All Bs are Cs, so all As are Cs.
Case: All As are Bs; therefore it is not the case that some As are not
Bs.
All As are Bs; so Some As are Bs.
Conversion:
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Case: All As are Bs; so All Bs are As.
Categorical Statements
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Statements in categorical logic have a specific
structure.
All human beings are mortal.
 Some four-legged creatures are gnus.
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They have the following structure:
quantifier + subject phrase + <link verb ‘be’> +
predicate phrase
 All categorical statements are structured like this.
 The link verb ‘be’ is called ‘copula’.
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Quantification
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Categorical statements are different from
statements in propositional logic: they have
quantifiers to quantify a statement.
Consider:
All human beings are mortal.
 Some people are rich.
 No pigs are able to fly.
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Here, all, some, and no are quantifiers.
Subject phrase and Predicate phrase
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Subject phrase and predicate phrase are
understood as classes, i.e. a set of objects that fall
in the phrase .
Human being: it is a class/set of all objects that are
human beings.
 Mortal: it is a class/set of all objects that are mortal.
 Socrates: this is an one-man class, which contains
only one person-Socrates.
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Ordinary language
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Sometimes the predicate phrases are not
explicitly referring to a class; but we can always
transform them to its corresponding class.
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Sometimes copula is not present either, and we can
add one.
Examples:
I love apples. == I am the one who loves apples
 Bees are angry == Bees are one of the things that
are angry.
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Four Kinds of Categorical
Statements
All S are P.
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It asserts that every member of class S is a member of
class P.
Some S are P.
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At least one member of S is also in the class P.
Note the use of the word ‘Some’: it does not imply that
there are more than one member of S is in P.
No S are P.
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No member of S is in the class P.
Some S are not P.
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At least one member of S is not in the class P.
Traditional Terminology
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In traditional logic, each of four kinds of
categorical statements is given a names.
A: All S are P. (Universal Affirmative)
E: No S are P. (Universal Negative)
I: Some S are P. (Particular Affirmative)
O: Some S are not P. (Particular Negative)
Further explanations
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There are two ways to characterize a categorical
statement:
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Quality: whether the statement confirms (a confirmative) or
negates (a negative);
Quantity: whether it has a universal (all) or a particular
quantifier (some).
Types of Categorical Statements
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A: all As are Bs: it is a Universal Confirmative.
E: no As are Bs: it is the same as “All As are not Bs”; so it is
a Universal Negative.
I and O statements can be similarly understood.
Translations
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In ordinary language, categorical statements are
often not put in a standard form. So translations
are often needed in order for us to have a
precise and accurate understanding of these
statements.
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The standard form is also important for us to
capture the argument relation between categorical
statements.
Missing Copula or Quantifier
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Copula or quantifiers are not present in these
statements:
Dogs love meat.
 Workers should get paid.
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Solution: add the missing parts (without
changing the meanings of the statements)
All dogs are the animals that love meat.
 All workers are the people who should get paid.
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Terms without Nouns
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Some statements may not have nouns as
predicates:
Roses are red;
 All ducks swim.
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Solution: replace them with a noun phrase or a
noun clause
All roses are red things.
 All ducks are animals that swim.
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Singular Statements
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What about statements with a singular subject term?
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G. W. Bush is a good president.
George loves Starbucks.
Rule: treat singular statement as A-statement.
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Singular term is an one-object class. Then it says about
everything in the class.
G. W. Bush is a good president == all people who are
identical with G. W. Bush is a good president.
George loves Starbucks==all people who is identical with
George love Starbucks.
Other expressions for the Universal
Quantifier
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Cases
Every soldier is a warrior.
 Each one of you is responsible.
 Whoever is a doctor earns a lot of money.
 Any tiger can be dangerous.
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These are all A-statements; these quantifiers are
the same as ‘all’.
Every soldier is a warrior.
 All soldiers are warriors.
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E-Statement
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Cases:
Nobody loves Ray.
 Nothing is better than pure love.
 None of the animals are alive.
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Rule: these are all E-statements. Treat these
quantifiers as ‘no’.
Nobody loves Ray.
 No people are the people who love Ray.
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I-Statement
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Cases
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Most students are honest.
Many people are retiring late.
A few dogs got killed.
At least one person is missing.
Almost all the cats have four legs.
There are government employees who are spies.
Rule: these are all I-statements. Treat all these
quantifiers as ‘some’.
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There is a significant difference between ‘most’ ‘many’ and
‘some’, but it is beyond our concern here.
O-Statements
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Cases:
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Not all the rich people are smart.
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There are government employees who are not
qualified.
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Some rich people are not smart.
Some government employees are not qualified.
Most Democrats are not in favor of the war.
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Some Democrats are not in favor of the war
Other Structures
Not all A are B ≠ No A are B.
Not all A are B:
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This is an O-Statement: some As are not Bs.
No A are B
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This is an E-Statements. No As are Bs.
Case:
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Not all Republicans support the war.
No Republicans support the war.
‘Only’ & ‘only if’
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‘Only’?
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‘only if’?
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Only rich people are invited.
Only if one studies logic one gets smart.
Translation rule:
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These statements are A-statements. The term after ‘only’ or
‘only if’ is predicate term.
Only rich people are invited == all invited people are rich
ones.
Only if one studies logic one gets smart. == all smart people
are those who study logic.
‘The Only’
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Case:
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Translation rule:
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The only guests invited are boys.
Cockroaches are the only survivors.
These are A-statements; the term that occurs after ‘the only’
is the subject term.
Translation:
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The only guests invited are boys == all guests invited are
boys.
Cockroaches are the only survivors == all survivors are
cockroaches.
Try these exercises in the book!
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Page 263, Ex. 7.2: Questions No. 4, 6, 8, 10, 14,
15.
Page 263-4, Ex. 7.3: No. 10, 13, 14, 17.
Ex. 7.2
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#4 People who whisper lie.
All people who whisper are people who lie.
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#6 Only if something has a back beat is it a rock-androll song.
All rock-and-roll songs are things with a back beat.
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A-statement
A-statement (note the predicate and the subject)
Only, only if: what follows are predicates of an A-statement
The only: what follows are subjects of an A-statement
#8: Nothing that is a snake is a mammal.
No snakes are mammals. E-statement.
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#10: The only good human is a dead human.
All good humans are dead humans.
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#14: There is no excellence without difficulty.
No excellence is without difficulty.
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E-statement.
Or: All excellent things are things with difficulty.
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A-statement.
A-statement
#15: Jonathan is not a very brave pilot.
No people who are identical to Jonathan are very brave pilots.
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E-statement.
Is this an A-statement? Not really.
A statement vs. E-statement
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A statement: All S are P.
E-statement: No S are P.
What about?
All S are not P:
 No S are not P:
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E-statement
A-statement
Why? Venn diagrams show they are so.
Compare:
Some S are P.
 Some S are not P.
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Ex. 7.3: No. 10, 13, 14, 17
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10: People who love only once in their lives are
shallow people. (Oscar Wilde)
All the people who love only once in their lives
are shallow people.
 A-statement
13: Many socialists are not communists.
Some socialists are not communists.
 O-statement
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14: All prejudices may be traced to the intestines.
All prejudices are things that may be traced to the
intestines.
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17: He that is born to be hanged shall never be
drowned.
All people who are born to be hanged are people who
shall never be drowned.
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A-statement.
A -statement
No people who are born to be hanged are the people
who shall be drowned
 E-statement
Quiz
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In this quiz you will be give a set of categorical
statements in English, and you are asked to
translated them into standard form of a
categorical statement, and indicate its type (A, E,
I, O).
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It is similar to the exercises we did in the previous
slides.